13.1-4. Temperature Distribution in a Hollow Sphere. Derive Eq. (13.1-18) for the steady- state conduction of heat in a hollow sphere. Also, derive an equation that shows that the temperature varies hyperbolically with the radius T- T Ans. sqty Juo e 12 T, T
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- 2.2 A small dam, which is idealized by a large slab 1.2 m thick, is to be completely poured in a short Period of time. The hydration of the concrete results in the equivalent of a distributed source of constant strength of 100 W/m3. If both dam surfaces are at 16°C, determine the maximum temperature to which the concrete will be subjected, assuming steady-state conditions. The thermal conductivity of the wet concrete can be taken as 0.84 W/m K.2.51 Determine by means of a flux plot the temperatures and heat flow per unit depth in the ribbed insulation shown in the accompanying sketch.1.10 A heat flux meter at the outer (cold) wall of a concrete building indicates that the heat loss through a wall of 10-cm thickness is . If a thermocouple at the inner surface of the wall indicates a temperature of 22°C while another at the outer surface shows 6°C, calculate the thermal conductivity of the concrete and compare your result with the value in Appendix 2, Table 11.
- 2.30 An electrical heater capable of generating 10,000 W is to be designed. The heating element is to be a stainless steel wire having an electrical resistivity of ohm-centimeter. The operating temperature of the stainless steel is to be no more than 1260°C. The heat transfer coefficient at the outer surface is expected to be no less than in a medium whose maximum temperature is 93°C. A transformer capable of delivering current at 9 and 12 V is available. Determine a suitable size for the wire, the current required, and discuss what effect a reduction in the heat transfer coefficient would have. (Hint: Demonstrate first that the temperature drop between the center and the surface of the wire is independent of the wire diameter, and determine its value.)1.4 To measure thermal conductivity, two similar 1-cm-thick specimens are placed in the apparatus shown in the accompanying sketch. Electric current is supplied to the guard heater, and a wattmeter shows that the power dissipation is 10 W. Thermocouples attached to the warmer and to the cooler surfaces show temperatures of 322 and 300 K, respectively. Calculate the thermal conductivity of the material at the mean temperature in W/m K. Problem 1.43.5 In a ball-bearing production facility, steel balls that are each of 15 mm in diameter are annealed by first heating them to 870°C and then slowly cooling in air to 125°C. If the cooling air stream temperature is 60°C, and it has a convective heat transfer coefficient of , determine the time required for the cooling.
- 2.44 To determine the thermal conductivity of a long, solid 2.5-cm-diameter rod, one half of the rod was inserted into a furnace while the other half was projecting into air at . After steady state had been reached, the temperatures at two points 7.6 cm apart were measured and found to be and , respectively. The heat transfer coefficient over the surface of the rod exposed to the air was estimated to be 22.7 . What is thermal conductivity of the rod?3.14 A thin-wall cylindrical vessel (1 m in diameter) is filled to a depth of 1.2 m with water at an initial temperature of 15°C. The water is well stirred by a mechanical agitator. Estimate the time required to heat the water to 50°C if the tank is suddenly immersed in oil at 105°C. The overall heat transfer coefficient between the oil and the water is , and the effective heat transfer surface is .Heat is generated uniformly in the fuel rod of a nuclear reactor. The rod has a long, hollow cylindrical shape with its inner and outer surfaces at temperatures of TiandTo, respectively. Derive an expression for the temperature distribution.
- Water at a temperature of 77C is to be evaporated slowly in a vessel. The water is in a low-pressure container surrounded by steam as shown in the sketch below. The steam is condensing at 107C. The overall heat transfer coefficient between the water and the steam is 1100W/m2K. Calculate the surface area of the container that is required to evaporate water at a rate of 0.01 kg/s. Problem 1.162.15 Suppose that a pipe carrying a hot fluid with an external temperature of and outer radius is to be insulated with an insulation material of thermal conductivity k and outer radius . Show that if the convection heat transfer coefficient on the outside of the insulation is and the environmental temperature is , the addition of insulation actually increases the rate of heat loss if , and the maximum heat loss occurs when . This radius, is often called the critical radius.A cylindrical liquid oxygen (LOX) tank has a diameter of 1.22 m, a length of 6.1 m, and hemispherical ends. The boiling point of LOX is -179.4C. An insulation is sought that will reduce the boil-off rate in the steady state to no more than 11.3 kg/h. The heat of vaporization of LOX is 214 kJ/kg. If the thickness of this insulation is to be no more than 7.5 cm, what would the value of its thermal conductivity have to be?